Orthogonality relations of Crouzeix-Raviart and Raviart-Thomas finite element spaces

Identities that relate projections of Raviart-Thomas finite element vector fields to discrete gradients of Crouzeix-Raviart finite element functions are derived under general conditions. Various implications such as discrete convex duality results and a characterization of the image of the projection of the Crouzeix-Ravaiart space onto elementwise constant functions are deduced.

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1. Introduction

Recent developments in the numerical analysis of total variation regularized and related nonsmooth minimization problems show that nonconforming and discontinuous finite element methods lead to optimal convergence rates under suitable regularity conditions [ChaPoc19-pre, Bart20a-pre, Bart20b-pre]. This is in contrast to standard conforming methods which often perform suboptimally [BaNoSa15]

. A key ingredient in the derivation of quasi-optimal error estimates are discrete convex duality results which exploit relations between Crouzeix–Raviart and Raviart–Thomas finite element spaces introduced in

[CroRav73] and [RavTho77]. In particular, assume that is a bounded Lipschitz domain with a partitioning of the boundary into subsets , and let be a regular triangulation of . For a function and a vector field we then have the integration-by-parts formula

 ∫Ω∇hvh⋅yhdx=−∫Ωvhdivyhdx.

Important aspects here are that despite the possible discontinuity of and no terms occur that are related to interelement sides and that the vector field and the function can be replaced by their elementwise averages on the left- and right-hand side, respectively. In combination with Fenchel’s inequality this implies a weak discrete duality relation.

The validity of a strong discrete duality principle has been established in [ChaPoc19-pre, Bart20a-pre] under certain differentiability or more generally approximability properties of minimization problems using the orthogonality relation

 (1) (ΠhRT0N(Th))⊥=∇h(kerΠh|S1,crD(Th)),

within the space of piecewise constant vector fields equipped with the inner product and with and denoting the elementwise application of the gradient and orthogonal projection onto , respectively, denotes the kernel of an operator. The identity implies that if a vector field satisfies

 ∫Ωwh⋅∇hvhdx=0

for all with then there exists a vector field such that

 wh=Πhzh.

Note that this is a stronger implication then the well known result that if is orthogonal to discrete gradients of all Crouzeix–Raviart functions then it belongs to the Raviart–Thomas finite element space. Although strong duality is not required in the error analysis, it reveals a compatibility property of discretizations and indicates optimality of estimates. Moreover, it is related to postprocessing procedures that provide the solution of computationally expensive discretized dual problems via simple postprocessing procedures of numerical solutions of less expensive primal problems, cf. [Mari85, ArnBre85, CarLiu15, Bart20a-pre].

The proof of (1) given in [ChaPoc19-pre] makes use of a discrete Poincaré lemma which is valid if the Dirichlet boundary is empty or if and is connected. In this note we show that (1) can be established for general boundary partitions by avoiding the use of the discrete Poincaré lemma. The new proof is based on the surjectivity property of the discrete divergence operator

 div:RT0N(Th)→L0(Th).

This is a fundamental property for the use of the Raviart–Thomas method for discretizing saddle-point problems, cf. [RavTho77, BoBrFo13-book]

. It is an elementary consequence of a projection property of a quasi-interpolation operator

and the surjectivity of the divergence operator onto the space .

Our arguments also provide a dual version of the orthogonality relation (1) which states that

 (2) div(kerΠh|RT0N(Th))=(ΠhS1,crD(Th))⊥.

Unless we have that the left-hand side is trivial and hence the identity yields that

 ΠhS1,crD(Th)=L0(Th),

i.e., that the projection of Crouzeix–Raviart functions onto elementwise constant functions is a surjection. If then depending on the triangulation both equality or strict inclusion occur. This observation reveals that the discretizations of total-variation regularized problems devised in [ChaPoc19-pre, Bart20a-pre] can be seen as discretizations using elementwise constant functions with suitable nonconforming discretizations of the total variation functional.

The most important consequence of (2) is the strong duality relation for the discrete primal problem defined by minimizing the functional

 Ih(uh)=∫Ωϕ(∇huh)+ψh(x,Πhuh)dx

in the space and the discrete dual problem consisting in maximizing the functional

 Dh(zh)=−∫Ωϕ∗(Πhzh)+ψ∗h(x,divzh)dx

in the space . The functions and are suitable convex functions with convex conjugates and , we refer the reader to [Bart20a-pre] for details.

This article is organized as follows. In Section 2 we define the required finite element spaces along with certain projection operators. Our main results are contained in Section 3, where we prove the identities (1) and (2) and deduce various corollaries. In the Appendix A we provide a proof of the discrete Poincaré lemma that leads to an alternative proof of the main result under certain restrictions.

2. Preliminaries

2.1. Triangulations

Throughout what follows we let be a sequence of regular triangulations of the bounded polyhedral Lipschitz domain , cf. [BreSco08-book, Ciar78-book]. We let denote the set of polynomials of maximal total degree on and define the set of elementwise polynomial functions or vector fields

 Lk(Th)ℓ={wh∈L∞(Ω;Rℓ):wh|T∈Pk(T) for all T∈Th}.

The parameter refers to the maximal mesh-size of the triangulation . The set of sides of elements is denoted by . We let and denote the midpoints (barycenters) of sides and elements, respectively. The projection onto piecewise constant functions or vector fields is denoted by

 Πh:L1(Ω;Rℓ)→L0(Th)ℓ.

For we have for all . We repeatedly use that is self-adjoint, i.e.,

 (Πhv,w)=(v,Πhw)

for all with the inner product .

2.2. Crouzeix–Raviart finite elements

The Crouzeix–Raviart finite element space introduced in [CroRav73] consists of piecewise affine functions that are continuous at the midpoints of sides of elements, i.e.,

 S1,cr(Th)={vh∈L1(Th):vh continuous in xS for all S∈Sh}.

The elementwise application of the gradient operator to a function defines an elementwise constant vector field via

 ∇hvh|T=∇(vh|T)

for all . For we have . Functions vanishing at midpoints of boundary sides on are contained in

 S1,crD(Th)={vh∈S1,cr(Th):vh(xS)=0 for all S∈Sh with S⊂ΓD}.

A basis of the space is given by the functions , , satisfying the Kronecker property

 φS(xS′)=δS,S′

for all . The function vanishes on elements that do not contain the side and is continuous with value 1 along . A quasi-interpolation operator is for defined via

 Icrv=∑S∈ShvSφS,vS=|S|−1∫Svds,

We have that preserves averages of gradients, i.e.,

 ∇hIcrv=Πh∇v,

which follows from an integration by parts, cf. [BoBrFo13-book, Bart16-book].

2.3. Raviart–Thomas finite elements

The Raviart–Thomas finite element space of [RavTho77] is defined as

 RT0(Th)={yh∈H(div;Ω):yh|T(x)=aT+bT(x−xT),aT∈Rd,bT∈R for all T∈Th},

where is the set of vector fields whose distributional divergence belongs to . Vector fields in have continuous constant normal components on element sides. The subset of vector fields with vanishing normal component on the Neumann boundary is defined as

 RT0N(Th)={yh∈RT0(Th):yh⋅n=0 on ΓN},

where is the outer unit normal on . A basis of the space is given by vector fields associated with sides . Each vector field is supported on adjacent elements with

 (3) ψS(x)=±|S|d!|T±|(zS,T±−x)

for with opposite vertex to . We have the Kronecker property

 ψS|S′⋅nS′=δS,S′

for all sides with unit normal vector , if we assume that points from into . A quasi-interpolation operator is for vector fields given by

 IRTz=∑S∈ShzSψS,zS=|S|−1∫Sz⋅nSds.

For the operator we have the projection property

 divIRTz=Πhdivz,

which is a consequence of an integration by parts, cf. [BoBrFo13-book, Bart16-book]. This identity implies that the divergence operator defines a surjection from into , provided that constants are eliminated from if .

2.4. Integration by parts

An elementwise integration by parts implies that for and we have the integration-by-parts formula

 (4) ∫Ω∇hvh⋅yhdx+∫Ωvhdivyhdx=∫∂Ωvhyh⋅nds.

Here we used that has continuous constant normal components on inner element sides and that jumps of have vanishing integral mean. If an elementwise constant vector field satisfies

 ∫Ωwh⋅∇hvhdx=0

for all then by choosing for one finds that its normal components are continuous on inner element sides and vanish on the , so that . We thus have the decomposition

 L0(Th)d=ker(div|RT0N(Th))⊕∇hS1,crD(Th),

where we used that .

3. Orthogonality relations

The following identities and in particular their proofs and corollaries are the main contributions of this article.

Theorem 3.1 (Orthogonality relations).

Within the sets of elementwise constant vector fields and functions equipped with the inner product we have

 (ΠhRT0N(Th))⊥=∇h(kerΠh|S1,crD(Th)),div(kerΠh|RT0N(Th))=(ΠhS1,crD(Th))⊥.
Proof.

(i) The integration-by-parts formula (4) implies

 (∇hvh,Πhyh)=−(vh,divyh)=−(Πhvh,divyh)=0

if and hence . To prove the converse inclusion let be orthogonal to . We show that there exists with . For this, let and be the uniquely defined function with

 (Πhrh,Πhvh)=(yh,∇hvh)

for all . The identity holds for all since is orthogonal to discrete gradients of functions with . In particular, is orthogonal to constant functions if . We choose with and verify that

 (yh−zh,∇hvh)=(Πhrh,Πhvh)+(divzh,vh)=0

for all . We next define for all and note that

 (˜yh−zh,∇hvh)=(yh−zh,∇hvh)=0

for all . Since is elementwise constant, it follows that and in particular . By definition of we have which proves the first asserted identity.
(ii) For the second statement we first note that if for then

 (Πhvh,divyh)=(vh,divyh)=−(∇hvh,Πhyh)=0

for all and hence . It remains to show that

 (ΠhS1,crD(Th))⊥⊂div(kerΠh|RT0N(Th)).

If is orthogonal to we choose with and note that

 (Πhzh,∇hvh)=(zh,∇hvh)=−(wh,vh)=−(wh,Πhvh)=0

for all . This implies that and hence also . Since and we deduce the second identity. ∎

An implication is a surjectivity property of the mapping if .

Corollary 3.2 (Surjectivity).

If then we have

 ΠhS1,crD(Th)=L0(Th).

Otherwise, the subspace has codimension at most one.

Proof.

(i) From Theorem 3.1 we deduce that the asserted identity holds if and only if . Since

 kerΠh|RT0N(Th)={yh∈RT0N(Th):yh|T=bT(x−xT) f.a. T∈Th},

the latter condition is equivalent to . Let such that a side belongs to , i.e., we have , where is the vertex of opposite to the side , and with . If then it follows that for since the vectors are linearly independent. Starting from this element we may successively consider neighboring elements to deduce that for all .
(ii) If we may argue as in (i) by removing one side from , define , and using the larger space . We then have . The difference is trivial if and only if belongs to . ∎

The following examples show that both equality or strict inequality can occur if .

Examples 3.3.

(i) Let , , , . Then while .
(ii) Let be a triangulation consisting of the subtriangles obtained by connecting the vertices of a macro triangle with its midpoint . Let and . We then have .

The second implication concerns discrete versions of convex duality relations. We let

 ϕ∗(s)=supr∈Rℓs⋅r−ϕ(r)

be the convex conjugate of a given convex function .

Corollary 3.4 (Convex conjugation).

Let and be convex. We then have

 inf{∫Ωϕ(∇huh)dx:uh∈S1,crD(Th),Πhuh=¯¯¯uh}≥sup{−∫Ωϕ∗(Πhzh)dx−(¯¯¯uh,divzh):zh∈RT0N(Th)}.

If and the infimum is finite then equality holds.

Proof.

An integration by parts and Fenchel’s inequality show that

 (5) −(Πhuh,divzh)=(∇huh,Πhzh)≤ϕ(∇huh)+ϕ∗(Πhzh).

This implies that the left-hand side is an upper bound for the right-hand side. If is differentiable is optimal in the infimum then we have the optimality condition

 ∫Ωϕ′(∇huh)⋅∇hvhdx=0

for all with . Theorem 3.1 yields that for some . This identity implies equality in (5) and hence

 ∫Ωϕ(∇huh)dx=−∫Ωϕ∗(Πhzh)dx−(¯¯¯uh,divzh)

so that the asserted equality follows. ∎

Remark 3.5.

For nondifferentiable functions , the strong duality relation can be established if there exists a sequence of continuously differentiable functions such that the corresponding discrete primal and dual problems and are -convergent to and as , respectively. An example is the approximation of by functions for .

With the conjugation formula we obtain a canonical definition of a discrete dual variational problem.

Corollary 3.6 (Discrete duality).

Assume that is convex and is elementwise constant in the first argument and convex with respect to the second argument. For and define

 Ih(uh)=∫Ωϕ(∇huh)+ψh(x,Πhuh)dx,Dh(zh)=−∫Ωϕ∗(Πhzh)+ψ∗h(x,divzh)dx.

We then have

 infuh∈S1,crD(Th)Ih(uh)≥supzh∈RT0N(Th)Dh(zh).
Proof.

Using the result of Corollary 3.4 and exchanging the order of the extrema we find that

 infuhIh(uh)≥infuhsupzh−∫Ωϕ∗(Πhzh)dx−(Πhuh,divzh)+∫Ωψh(x,Πhuh)dx≥supzh−∫Ωϕ∗(Πhzh)dx+infuh−(Πhuh,divzh)+∫Ωψh(x,Πhuh)dx=supzh−∫Ωϕ∗(Πhzh)dx−supuh(Πhuh,divzh)−∫Ωψh(x,Πhuh)dx≥supzh−∫Ωϕ∗(Πhzh)dx−∫Ωψ∗h(x,Πhuh)dx=supzhDh(zh).

This proves the asserted inequality. ∎

The fourth implication concerns the postprocessing of solutions of the primal problem to obtain a solution of the dual problem. This also implies a strong discrete duality relation.

Corollary 3.7 (Strong discrete duality).

In addition to the conditions of Corollary 3.6 assume that and is finite and differentiable with respect to the second argument. If is minimal for then the vector field

 zh=ϕ′(∇huh)+ψ′h(x,Πhuh)d−1(1−Πh)id

is maximal for with .

Proof.

The optimal solves the optimality condition

 (6) ∫Ωϕ′(∇huh)⋅∇hvh+ψ′h(⋅,Πhuh)Πhvhdx=0

for all . By restricting to functions satisfying we deduce with Theorem 3.1 that there exists with

 Πhzh=ϕ′(∇huh).

The optimality condition (6) implies that . Hence, satisfies the asserted identity. With the resulting Fenchel identities

 ∇huh⋅Πhzh=ϕ(∇huh)+ϕ∗(Πhzh),Πhuh⋅divzh=ψh(⋅,Πhuh)+ψ∗h(⋅,divzh),

and by choosing in (6) we find that

 Ih(uh)=Dh(zh)

which in view of the weak duality relation implies that is optimal. ∎

Appendix A Discrete Poincaré lemma

For completeness we provide a derivation of (1) based on a discrete Poincaré lemma. We say that is connnected if its relative interior has at most one connectivity component.

Proposition A.1 (Discrete Poincaré lemma).

Assume that or and is connected. A vector field satisfies for a function if and only if

 ∫Ωwh⋅yhdx=0

for all with .

Proof.

If then the orthogonality relation follows from the integration-by-parts formula (4). Conversely, if is orthogonal to vector fields with vanishing divergence then we can construct a function by integrating along a path connecting midpoints of sides, i.e., choosing a side at which some value is assigned to , e.g., . If then we choose . The values at other sides are obtained via

 vh(xS)=vh(xS0)+J∑j=1wh|Tj⋅(xSj−xSj−1),

where is a chain of (unique) elements connecting with via the shared sides , i.e., for . To see that this is well defined it suffices to show that for every closed path with the sum equals zero. To verify this we define the Raviart–Thomas vector field

 yh=J∑j=1(d−1)!|Sj|ψSj,